3 SemicircleDistribution
3.1 Semicircle Probability Density Function
The function \(\mathrm{sc} : \mathbb {R} \times \mathbb {R}_{\geq 0} \times \mathbb {R} \rightarrow \mathbb {R}\) defined by
is called the probability density function (pdf) of the semicircle distribution.
Given a mean \(\mu \in \mathbb {R}\) and a variance \(v \in \mathbb {R}_{\geq 0}\), the pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution with mean \(\mu \) and variance \(v\) is given by
If the variance \(v\) is given to be zero, then the pdf of the semicircle distribution is the function that is identically zero.
By Definition 3.1.1, the square root of a nonpositive number is defined to be zero. Hence, the pdf with a zero variance must be the function that is identically zero.
The pdf of the semicircle distribution is always nonnegative.
By Definition 3.1.1, the square root of a nonpositive number is defined to be zero. Furthermore, the variance is always assumed to be nonnegative. Therefore, since the fractional term and the square root term are always nonnegative, we conclude the pdf is always nonnegative.
Given a mean \(\mu \in \mathbb {R}\) and a variance \(v \in \mathbb {R}_{\geq 0}\), the pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution with mean \(\mu \) and variance \(v\) is measurable.
Given a mean \(\mu \in \mathbb {R}\) and a variance \(v \in \mathbb {R}_{\geq 0}\), the pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution with mean \(\mu \) and variance \(v\) is strongly measurable.
By Lemma 3.1.5, we know the pdf \(\mathrm{sc}\) with fixed mean \(\mu \) and variance \(v\) is measurable. Since \(\mathbb {R}\) is equipped with a second countable topology, the fact that \(\mathrm{sc}\) with fixed mean \(\mu \) and variance \(v\) implies \(\mathrm{sc}\) is strongly measurable.
Given a mean \(\mu \in \mathbb {R}\) and a variance \(v \in \mathbb {R}_{\geq 0}\), the pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution with mean \(\mu \) and variance \(v\) is integrable.
Given a mean \(\mu \in \mathbb {R}\) and a nonzero variance \(v \in \mathbb {R}_{{\gt} 0}\), the lower Lebesgue integral of the p.d.f. \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution with mean \(\mu \) and variance \(v\) equals \(1\).
Given a mean \(\mu \in \mathbb {R}\) and a nonzero variance \(v \in \mathbb {R}_{{\gt} 0}\), the integral of the pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution with mean \(\mu \) and variance \(v\) equals \(1\).
For any pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution, the following relation is satisfied:
for any \(\mu \in \mathbb {R}\), \(v \in \mathbb {R}_{\geq 0}\), and \(x,y \in \mathbb {R}\).
Expanding Definition 3.1.1 gives
For any pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution, the following relation is satisfied:
for any \(\mu \in \mathbb {R}\), \(v \in \mathbb {R}_{\geq 0}\), and \(x,y \in \mathbb {R}\).
Expanding Definition 3.1.1 gives
For any pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution, the following relation is satisfied:
for any \(\mu \in \mathbb {R}\), \(v \in \mathbb {R}_{\geq 0}\), \(x \in \mathbb {R}\), and nonzero \(c \in \mathbb {R}\).
For any pdf \(\mathrm{sc} : \mathbb {R} \rightarrow \mathbb {R}\) of the semicircle distribution, the following relation is satisfied:
for any \(\mu \in \mathbb {R}\), \(v \in \mathbb {R}_{\geq 0}\), \(x \in \mathbb {R}\), and nonzero \(c \in \mathbb {R}\).
Expanding Definition 3.1.1 gives
3.2 To Extended Nonnegative Reals
Let \(f : \mathbb {R} \times \mathbb {R}_{\geq 0} \times \mathbb {R} \to \mathbb {R}\) denote the real-valued semicircle density defined in Definition 3.1.1. Define the function \(h: \mathbb {R} \to \mathbb {R} \cup \{ \infty \} \) as follows:
Then we define the function \( g : \mathbb {R} \times \mathbb {R}_{\geq 0} \times \mathbb {R} \to [0,\infty ] \subseteq \overline{\mathbb {R}}_{\ge 0}\) by:
For all \(\mu \in \mathbb {R} , v \in \mathbb {R}_{\geq 0}\), the extended pdf. \( \bar{sc} : \mathbb {R} \to [0,\infty ]\) satisfies:
If the variance \(v\) is zero, then the extended pdf. is identically zero:
This follows immediately from the definition of \(\bar{sc}\) as \(h(sc(\mu ,0,x))\), and the fact that \(sc(\mu ,0,x) = 0\) from Lemma 3.1.3.
Let \( \mu \in \mathbb {R} , v \in \mathbb {R}_{\ge 0}\), and \(x \in \mathbb {R} \). Then the real value recovered from the extended semicircle PDF satisfies:
Since \(sc(\mu , v, x) \ge 0\), we have \(h(sc(\mu , v, x)) = sc(\mu , v, x)\), and thus
Therefore,
as desired.
If \(v {\gt} 0\), then for all \(\mu , x \in \mathbb {R}\), the extended pdf is nonnegative:
This is immediate from the definition of \(\bar{sc}\) as \(h(sc(\mu ,v,x))\) and the nonnegativity of \(sc\) (Lemma 3.1.4).
For all \(\mu , x \in \mathbb {R}\), and \(v \in \mathbb {R}_{\ge 0}\), we have:
Since \(sc(\mu ,v,x) \in \mathbb {R}_{\ge 0}\), we have \(\bar{sc}(\mu ,v,x) = h(sc(\mu ,v,x)) {\lt} \infty \).
For all \( \mu , x \in \mathbb {R} \), and \( v \in \mathbb {R}_{\ge 0} \), the extended pdf is finite:
Let \( \mu \in \mathbb {R} \) and \( v \in \mathbb {R}_{{\gt} 0} \). Then the support of the extended pdf is
sorry
The function \( x \mapsto \bar{sc}(\mu ,v,x) \) is measurable for all \( \mu \in \mathbb {R} \), \( v \in \mathbb {R}_{\ge 0} \).
Since \(h\) is measurable, and \(h\) is a measurable map \( \mathbb {R}_{\ge 0} \to \overline{\mathbb {R}}_{\ge 0} \), their composition is measurable.
If \(v {\gt} 0\), then the total integral of \(\bar{sc}\) with respect to Lebesgue measure is 1:
This follows from the equality:
using Lemma 3.1.8.
3.3 Semicircle Distribution
The semicircle distribution with mean \(\mu \) and variance \(v\), denoted \(\sigma (\mu , v)\), is the Dirac delta at μ if v = 0; otherwise, it’s the measure with density with respect to the Lebesgue measure given by the semicircle pdf.
If \(v \neq 0\), then the definition the semicircle distribution is defined as the measure with density with respect to the Lebesgue measure given by the semicircle pdf.
Follows directly from definition of semicircle distribution.
If the variance is 0, then the semicircle distribution is exactly the Dirac measure at \(\mu \).
Follows directly from definition of semicircle distribution.
For all \(\mu \in \mathbb {R}\) and \(v \in \mathbb {R}_{\ge 0}\), semicircleReal is a probability measure.
If \(v {\gt} 0\), then the semicircle distribution has no atoms.
For a semicircle measure with mean \(\mu \) and variance \(v {\gt} 0\), the measure of any measurable set \(s\) equals the Lebesgue integral over \(s\) of the semicircle probability density function at x.
For any mean \(\mu \in \mathbb {R}\), any variance \(v {\gt} 0\), and any measurable set \(s\) of real numbers, the semicircle distribution measure of the set \(s\) equals the extended nonnegative real number version (ENNReal.ofReal) of the integral of the semicircle probability density function over s.
For a semicircle distribution with mean \(\mu \) and variance \(v {\gt} 0\), the measure \(\sigma (\mu , v)\) is absolutely continuous with respect to the Lebesgue measure.
The Radon–Nikodym derivative of the semicircle measure \(\sigma (\mu , v)\) with respect to the Lebesgue measure is almost everywhere equal to the semicircle probability density function \(SC(\mu \), \(v)\).
Let \( f : \mathbb {R} \to E \) be a function where \( E \) is a normed vector space over \(\mathbb {R}\). For the semicircle distribution with mean \(\mu \in \mathbb {R}\) and variance \( v {\gt} 0 \), we have:
3.4 Transformations
Given semicircular measure \(\sigma (\mu , v)\) with mean \(\mu \) and variance \(v\), then for any constant \(y \in \mathbb {R}\), the pushforward of \(\sigma (\mu , v)\) under the map \(x \mapsto x + y\) is again a semicircular measure \(\sigma (\mu + y, v)\).
Given semicircular measure \(\sigma (\mu , v)\) with mean \(\mu \) and variance \(v\), then for any constant \(y \in \mathbb {R}\), the pushforward of \(\sigma (\mu , v)\) under the map \(x \mapsto y + x\) is again a semicircular measure \(\sigma (\mu + y, v)\).
Obvious from commutativity between \(x + y\) and \(y + x\).
Given semicircular measure \(\sigma (\mu , v)\) with mean \(\mu \) and variance \(v\), then for any constant \(c \in \mathbb {R}\), the pushforward of \(\sigma (\mu , v)\) under the map \(x \mapsto c * x\) is again a semicircular measure \(\sigma (c\mu , c^2v)\).
Given semicircular measure \(\sigma \) with mean \(\mu \) and variance \(v\), then for any constant \(c \in \mathbb {R}\), the pushforward of \(\sigma (\mu , v)\) under the map \(x \mapsto x * c\) is again a semicircular measure \(\sigma (c\mu , c^2v)\).
Use commutativity between \(Xc\) and \(cX\).
Given semicircular measure \(\sigma (\mu , v)\) with mean \(\mu \) and variance \(v\), the pushforward of \(\sigma (\mu , v)\) under the map \(x \mapsto -x\) is again a semicircular measure \(\sigma (- \mu , v)\).
Special case of the multiplication by constant map with constant being \(-1\).
Given semicircular measure \(\sigma (\mu , v)\) with mean \(\mu \) and variance \(v\), then for any constant \(y \in \mathbb {R}\), the pushforward of \(\sigma (\mu , v)\) under the map \(x \mapsto x - y\) is again a semicircular measure \(\sigma ( \mu - y, v)\).
Use the map by addition of constant and substitute constant for its \(-1\) multiple.
Given semicircular measure \(\sigma (\mu , v)\) with mean \(\mu \) and variance \(v\), then for any constant \(y \in \mathbb {R}\), the pushforward of \(\sigma (\mu , v)\) under the map \(x \mapsto y - x\) is again a semicircular measure \(\sigma (y - \mu , v)\).
Given a real random variable \(X \sim \sigma (\mu , v)\) then for a constant \(y \in \mathbb {R}\), \(X + y \sim \sigma (\mu + y, v)\)
Given a real random variable \(X \sim \sigma (\mu , v)\) then for a constant \(y \in \mathbb {R}\), \(y + X \sim \sigma (\mu + y, v)\)
Given a real random variable \(X \sim \sigma (\mu , v)\), then for a constant \(c \in \mathbb {R}\), \(cX \sim \sigma (c\mu , c^2v)\)
Given a real random variable \(X \sim \sigma (\mu , v)\), then for a constant \(c \in \mathbb {R}\), \(Xc \sim \sigma (c \mu , c^2v)\)
3.5 Moments
If \(X \sim \sigma (\mu , v)\), then its expectation
If \(X \sim \sigma (\mu , v)\), then its variance \(Var(X) = v\)
The variance of a real semicircle distribution with parameter \((\mu , v)\) is its variance parameter \(v\)
All the moments of a real semicircle distribution are finite. That is, the identity is in \(L_p\) for all finite \(p\)
\(\mathbb {E}[(X - \mu )^{2n}] = v^n C_n \)
\(\mathbb {E}[(X - \mu )^{2n}] = v^n C_n \)
\(\mathbb {E}[(X - \mu )^{2n + 1}] = 0 \)
\(\mathbb {E}[(X - \mu )^{2n + 1}] = 0 \)